Chris Cowell-Shah writes "If David Foster Wallace can't explain infinity to us, nobody can. At least, that's what I told myself while anxiously waiting for his Everything and More: A Compact History of Infinity. The book promised to be an intellectual history of the mathematical concept of infinity, with heavy doses of history, math, and philosophy. And while it proves heavy going at times, I'm pleased to say that it delivers admirably on this promise." Read on for Cowell-Shah's lengthy review of Everything and More.
Wallace may be best known for his footnotes. Virtually everything he has written from his strange but mesmerizing novel Infinite Jest to his hilarious essay about cruise ships (the title work in A Supposedly Fun Thing I'll Never Do Again) to his oddly gripping treatise on the philosophy of dictionaries ("Tense Present" in the April 2001 issue of Harper's)has been liberally sprinkled with footnotes. And what footnotes! Many go on wild tangents. Some contain sub- or sub-sub-footnotes. Others are the length of novellas and could legitimately be reprinted separately from the main work. My point is that Wallace is, at heart, a scholar. He's interested in details. Combine this with an impressive background in math and logic (though he modestly claims a "medium-strong amateur interest in math and formal systems"), and he would seem to make the perfect tour guide for infinity, a concept that seems simple enough on the surface but which we generally suspect is far more complex than we realize.
The Book's Audience and Aims
DFW (an overabundance of abbreviations is one of his most prominent literary tics, and I'll follow his lead) calls Everything and More (henceforth EAM) "a piece of pop technical writing" for "readers who do not have pro-grade technical backgrounds." But the fact of the matter is that to truly follow and understand all (or even most) of his points, one needs to know a lot of math. I'm probably typical of the average reader of EAM: I went through the standard two-year calculus cycle in high school and college, and though most of it made sense at the time, these days I generally double-check my long division. While I've had a fair amount of tertiary-level logic and formal systems coursework while studying computer science and philosophy, even those subjects have grown fuzzy with time. But I am interested in this stuff, and I have the patience and analytical practice to wade through almost any argument or proof, so I would guess that my experience with EAM is pretty close to that of most Slashdot readers.
I should note that this work is really an extended essay rather than a book. Granted, it's a 300-page essay, but that's the term DFW insists on and it seems appropriate given the lack of chapters. The only structure is provided by relatively unhelpful section headers like "4b," and the work sometimes seems to lack convenient breaking points where the reader can pause to catch a breath. This is not a criticism, but the style of the essay does demand that the reader do his best to stay aware of where he is in the overall story of infinity and to be prepared for occasional gaps in the narrative thread. Read this like a math proof with lots of reviewing and re-reading and comparing of earlier and later claims and you should do all right. It's also worth pointing out that the word "history" in the essay's subtitle is important. DFW's goal is mainly to chronicle the ways in which early and not-so-early mathematicians approached the concept of infinity, rather than to explain what infinity is useful for or to give us new ways of thinking about the term. It will probably never have the same mass appeal that more colorful but less difficult books like James Gleick's Chaos or Douglas Hofstadter's Gödel, Escher, Bach have enjoyed, but this is not necessarily a bad thing. DFW has a narrower and more technical aim, and he generally hits his target.
What EAM CoversIt's probably better to think of the essay as a series of loosely related arguments and observations rather than a single mathematical story. With this in mind, let's go through some of the essay's sections. DFW opens by discussing what it means to engage in abstract thinking, then investigates the Principle of Induction (a crucial element in the development of infinity) and explains Euclid's proof that there is no largest prime. He (re-)introduces us to a number of high school math concepts, including such things as reductio ad absurdum proofs and the difference between modus ponens and modus tollens. This refresher is very helpful; I consider the book's opening section to be worth the price of admission all by itself.
Once we've got these preliminary concepts under our belt, DFW starts in with ancient Greek philosophers and mathematicians and begins constructing a vast pyramid of mathematical ideas that will eventually support Georg Cantor's notion of infinity at its tip. This nineteenth century German mathematician is the central figure in the book (to the extent that there is one), and DFW makes it clear early on that we're ultimately moving toward his ideas and his vision of infinity. A quick tour through the Greeks covers Pythagoras, Zeno's paradoxes, Aristotle's demolition thereof, and Plato's theory of forms. It's at this point that we are introduced to fascinating questions of mathematical epistemology and ontology, questions that were first mulled over by the Greeks but that remain largely unsettled even today. For example, what do we have to know in order to really know and understand a mathematical concept? And do numbers exist external to people (the Platonist view), or are they purely human constructs (the Intuitionist stance)?
DFW skips ahead to the seventeenth century, where he showcases Galileo's ideas in Two New Sciences and leads us through some of Newton's and Leibniz's independent contributions to the development of calculus. A wonderful discussion of the archetype of the insane mathematician follows (he makes the unsurprising claim that very few world-class mathematicians were terribly well-adjusted). He then chronicles the intellectual shift from math being thought of as empirical (grounded in actual things) to abstract (based on intangibles and relations between them). He does a good job of explaining how this abstraction works surprisingly well when applied to real problems (especially in engineering and physics). It's at this point (in section five of seven) that the mathematical heavy lifting begins. DFW delves deeper into calculus and the notion of limits, and significantly more mental energy is required if the reader wishes to follow carefully. Fortunately, close scrutiny isn't strictly required; even skimming this portion and picking up the thread again in section six yields good results. Now winding down, DFW introduces us to Fourier series and steps through Cantor's delightful diagonalization/denumeration proofs of the mind-warping claims that there are the same number of whole numbers as integers as rationals, and that the cardinality of the reals is larger than the cardinality of any of these other sets. A short excursis into set theory (like most of the rest of the book, it's thrown at us semi-haphazardly rather than being systematically presented), a longish explanation of Cantor's Continuum Hypothesis (a claim about the relations between the various "sizes" of infinity), and we're done. Exhausted and probably more than a little confused, but done.
EAM as a Mathematical History
There are two ways to judge EAM: as a work of mathematical history, and as a piece of English prose. I consider it adequately successful when viewed in the first light, but exemplary when viewed in the second. The math side of the book is probably best assessed by presenting a scattershot collection of my impressions, so let's start with those.
DFW is, in the main, aware of which portions will pose particular trouble for most readers. The prose is peppered with phrases like "Now you can probably feel a headache starting" or "Here's one of those places where it's simply impossible to tell whether what's just been said will make sense to a general reader," which are usually accompanied by extra explanations or illustrations to clarify the point just made. As an amateur mathematician, he may in fact be better at empathizing with his readers' difficulties than many professors are. It's hard to imagine the following passage (with its awestruck tone) appearing in a math textbook or college calculus lecture:
"Let's pause to consider the vertiginous levels of abstraction involved here. If the human CPU cannot apprehend or even really conceive of infinity, it is now apparently being asked to countenance an infinity of infinities, an infinite number of individual members of which are themselves not finitely expressible, all in an interval [0-1] so finite- and innocent-looking we use it in little kids' classrooms. All of which is just resoundingly weird."
As an example of how he leads readers around conceptual landmines, DFW is especially careful to steer us away from thinking that infinity is just a really large number. He invites us instead to consider it and its cousins to be entirely different sorts of objects than finite numbers, with very different properties. This segues into a first-rate explanation of how infinity-related paradoxes (including Zeno's famous arrow paradoxes) often go away, or more properly, cannot be meaningfully stated, once we stop treating infinity as a normal number or (for certain paradoxes) once we are clear on the difference between zero and nothing (or "not applicable"). These are nonobvious points that I had never considered, but which make perfect sense once carefully laid out and illustrated. Resolving these paradoxes turns out to be a crucial propelling force in the history of infinity: "By this point you've almost certainly discerned the Story of Infinity's overall dynamic, whereby certain paradoxes give rise to conceptual advances that can handle those original paradoxes but in turn give rise to new paradoxes, which then generate further conceptual advances, and so on."
Even if you're relatively uninterested in the concept of infinity, DFW's broad and extraordinarily literate survey of concepts like abstractness, limits, and induction make the book worthwhile. He does an especially good job of explaining the nature of abstraction and why abstract thinking is so difficult. The essay is replete with facts not directly relevant to infinity but still interesting to the scientifically inclined. For example, it turns out that 5 x 10^-44 seconds is generally acknowledged to be the smallest interval in which the normal concept of continuous time applies. And Bremermann's Limit (2.56 x 20^92) is the theoretical limit of the number of bits of information that could have been processed by the most powerful computer that could exist on earth (a computer with the mass of the earth that has existed as long as the earth). Problems involving more data than this (such can be found in statistical physics) are considered transcomputable, or not computable in any meaningful sense. These geeky trivia won't improve your life in any way, but it does stave off some of the inevitable monotony of pure math writing.
DFW has lots to say about mathematical pedagogy, including this harsh indictment:
"Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.... And, of course, rarely do students think to ask the formulas alone take so much work to 'understand' (i.e., to be able to solve problems correctly with), we often aren't aware that we don't understand them at all. That we end up not even knowing that we don't know is the really insidious part of most math classes."
Perhaps this concern for how math is taught leads him to focus his efforts strictly on core concepts rather than on the biographical gossip so often found in popular science writing. There are some fun notes about Cantor's personal life, but he's the only one who gets an extended biographical exegesis. This appears to be a conscious and reasoned decision on his part rather than an oversight ("Again, most of this personal stuff we're skipping") and I think it is a wise strategic move in that it keeps the reader's attention focused and undistracted.
As expected, this work does indeed swim in a sea of footnotes. DFW fans would be disappointed in anything less, but I have to confess to lightly skimming most of the footnotes after the first third of the essay. The most difficult or technical notes are marked "IYI" (for "If You're Interested"), but even the non-IYIspasm notes are full of some pretty thorny math; I found that they often proved more confusing than helpful. But readers more familiar with the subject matter might appreciate the additional historical context and suggestions for further exploration provided in the footnotes.
Overall, EAM is more successful at explaining the small problems, paradoxes, and steps in the creation of infinity than it is at stringing them all together into a coherent, easily followed, transparently structured whole. As an example of how well DFW deals with the small-scale issues, consider the following mind-boggling concept. It is of course impossible to fully wrap your mind around this sort of thing, but in the text that follows this quotation he does a sterling job of steering us toward comprehension:
"The Number Line is obviously infinitely long and comprises an infinity of points. Even so, there are just as many points in the interval 0-1 as there are on the whole Number Line. In fact, there are as many points in the interval .00000000001-.00000000002 as there are on the whole N. L. It also turns out that there are as many points in the above micro-interval (or one one-quadrillionth its size, if you like) as there are on a 2D plane, even if that plane is infinitely larger in any 3D shape, or in all of infinite 3D space itself."
On a similar theme, DFW gives a brilliantly simple and utterly convincing explanation of the cortex-withering claim that "the number of points in the closed-interval [0,1] is ultimately equal to the infinity of points on the whole Real Line stretching infinitely in both directions." But (and this is my biggest criticism) this essay really has to be read twice (or more) to get anywhere near full comprehension of the material. In this respect, it's a lot like an extended math proof or a very long philosophy paper. Repeated exposure makes it easier to follow the narrative flow and string the arguments and proofs together into a consistent thread of thought rather than isolated, self-contained concepts.
EAM as a Literary Work
As mentioned above, where EAM really shines is not as a math history, but rather as an example of pure writing. DFW's prose is clear, precise, witty, and creative. His literary idiosyncrasies may be an acquired taste, but once the reader gets used to the aesthetic feel of the essay it becomes hard not to consider it a stylistic tour de force. In many ways this doesn't feel like a math book at all. This is perhaps not surprising given that the author is, after all, mainly a novelist. He loves to make up words, use obscure words, or use common words in strange new ways. Your appreciation for this style will vary depending on your tolerance for neologisms like homodontic (meaning "having only a single type of tooth") or epistoschizoid (meaning, well, your guess is as good as mine), or unusual punctuation (Does he really need parentheses nested inside of other parentheses? As it turns out, yes.). But you also get exposed to real (and entertaining) words like clonic (involving muscle spasms -- nothing to do with clones), cephalalgia (headache), and peruke (the goofy hats worn by Dutch burghers in seventeenth century portraits). Sometimes it doesn't quite work (What does "We are now once again sort of out over our skis, chronologically speaking" mean? Anyone?), but the overall effect is a refreshing and fun change of pace from standard math or science writing.
DFW uses shorthand to an almost pathological degree. This takes some getting used to, but ultimately it makes his text wonderfully compact (OK, his sentences can be almost unparsably long, but he packs a ton of content into each one) and produces virtually no loss of comprehension. The text is sprinkled with abbreviations like "w/r/t" for "with respect to" and useful sentence fragments like "Meaning it doesn't seem logically impossible or anything," and "Goes on forever." This sort of shorthand is pervasive, but really is more of a help than a hindrance. They may not be everyone's cup of tea, but informal parenthetical phrases such as "they're reversed from the axes in the motion-type graphs you're apt to have had in school (long story; good reasons)" are usually very helpful and inject a nicely colloquial tone into a topic that is traditionally treated in the most formal (and dullest) of styles. Descriptions like this are what keep you going when the math gets tough:
"[T]he whole enterprise becom[es] such a towering baklava of abstractions and abstractions of abstractions that you pretty much have to pretend that everything you're manipulating is an actual, tangible thing or else you get so abstracted that you can't even sharpen your pencil, much less do any math."
Everything and More: A Compact History of Infinity is more or less what its title promises. I found it well worth the (not insignificant) effort to plow through, and I recommend it to anyone interested in mathematical and/or intellectual history, or to anyone curious about how difficult mathematical concepts can be discussed in a lively and engaging way. While most readers won't be able to follow all of the subtleties of his arguments with just one pass through the text, a single pass can still be well worthwhile. Those looking for an introduction to David Foster Wallace would be better served by one of his less difficult books (I especially recommend A Supposedly Fun Thing I'll Never Do Again), but for fans of his more technical, scholarly essays, this book is a welcome arrival.
Chris Cowell-Shah is a consultant with Accenture Technology Labs, the R&D branch of Accenture. His website is cowell-shah.com. You can purchase Everything and More: A Compact History of Infinity from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.